You should set a court date, write the letter, then appear in court and explain your situation to the judge. He or she may be sympathetic to you, especially since your record is clean otherwise. He or she can also give you an extended time to pay this tickets. You may want to find out if your county clerk does payment plans on traffic tickets. That may be the way to alleviate your financial stress. You would probably be best served if you hired an attorney to appear in court for you. The fee to go to court for a traffic ticket is usually pretty affordable.
You can write a letter, it certainly cannot hurt, I would also take a copy of the letter to court--especially if you are shy or you think that you may forget what it is you want to say. Traffic Magistrates DO have discretion to dimiss your ticket with a good driving record. WORST-Case scenario, you ask the judge to give you 60 or even 90 days to pay. There are a lot of traffic attorneys that stay in traffic court all day too. You could contact one in advance or just walk up to one in court and ask him/her if you can give them $20 bucks to walk up with you, but you do the talking and the attorney just fills in anything thats missing or that can help. I think the judge would respect that. I would do it if someone asked me. I'm already there anyway and it takes minute of my time. If they ask you for more than $30 tell them no.